(a)[4]
Use the substitution $u = x^2 - 3$ to show that $\int_{\sqrt{7}}^{\sqrt{12}} \frac{4x^3}{\sqrt{x^2 - 3}}\,dx = \int_{a}^{b} \frac{2u(u+3)}{\sqrt{u}}\,du$, with $a$ and $b$ as the values to determine.
(b)[4]
Hence, Calculate the exact value of $\int_{\sqrt{7}}^{\sqrt{12}} \frac{4x^3}{\sqrt{x^2 - 3}}\,dx$.